NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty y x)
(REPLACEMENT-MAP
(+ 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(p(x),y) -> p(+(y,x))
+(s(x),y) -> s(+(x,y))
+(0,y) -> y
p(s(x)) -> s(p(x))
s(p(x)) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty y x)
(REPLACEMENT-MAP
(+ 1, 2)
(p 1)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(p(x),y) -> p(+(y,x))
+(s(x),y) -> s(+(x,y))
+(0,y) -> y
p(s(x)) -> s(p(x))
s(p(x)) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 +(p(x),y) -> p(+(y,x))
 +(s(x),y) -> s(+(x,y))
 +(0,y) -> y
 p(s(x)) -> s(p(x))
 s(p(x)) -> x
-> Vars:
 y, x, y, x, y, x, x

-> Rlps:
 (rule: +(p(x),y) -> p(+(y,x)), id: 1, possubterms: +(p(x),y)->[], p(x)->[1])
 (rule: +(s(x),y) -> s(+(x,y)), id: 2, possubterms: +(s(x),y)->[], s(x)->[1])
 (rule: +(0,y) -> y, id: 3, possubterms: +(0,y)->[], 0->[1])
 (rule: p(s(x)) -> s(p(x)), id: 4, possubterms: p(s(x))->[], s(x)->[1])
 (rule: s(p(x)) -> x, id: 5, possubterms: s(p(x))->[], p(x)->[1])

-> Unifications:
(R1 unifies with R4 at p: [1], l: +(p(x),y), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: p(+(y,x)), r': s(p(x')))
(R2 unifies with R5 at p: [1], l: +(s(x),y), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: s(+(x,y)), r': x')
(R4 unifies with R5 at p: [1], l: p(s(x)), lp: s(x), sig: {x -> p(x')}, l': s(p(x')), r: s(p(x)), r': x')
(R5 unifies with R4 at p: [1], l: s(p(x)), lp: p(x), sig: {x -> s(x')}, l': p(s(x')), r: x, r': s(p(x')))

-> Critical pairs info:
<+(s(p(x')),y),p(+(y,s(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N1
<p(x'),s(p(p(x')))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<s(s(p(x'))),s(x')>   =>  Not trivial, Not overlay, Proper, NW0, N3
<+(x',y),s(+(p(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N4

-> Problem conclusions:
 Left linear, Right linear, Linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a TRS 


Problem 1: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: +(s(p(x')),y)
Nodes: [0,1,2,3,4]
Edges: [(0,1),(0,2),(1,3),(3,4)]
ID: 0 => ('+(s(p(x')),y)', D0)
ID: 1 => ('s(+(p(x'),y))', D1, R2, P[], S{x8 -> y, x9 -> p(x')}), NR: 's(+(p(x'),y))'
ID: 2 => ('+(x',y)', D1, R5, P[1], S{x12 -> x'}), NR: 'x''
ID: 3 => ('s(p(+(y,x')))', D2, R1, P[1], S{x6 -> y, x7 -> x'}), NR: 'p(+(y,x'))'
ID: 4 => ('+(y,x')', D3, R5, P[], S{x12 -> +(y,x')}), NR: '+(y,x')'
t: p(+(y,s(x')))
Nodes: [0]
Edges: []
ID: 0 => ('p(+(y,s(x')))', D0)
+(s(p(x')),y) ->* no union *<- p(+(y,s(x')))
"Not joinable"

The problem is not confluent.